# Two players alternately flip biased coin. What is bias of coin?

Two players, A and B, alternately and independently flip a biased coin and the first player to get a head wins. Assume player A flips first. Player A wins the game 12/23 times. If the coin is biased, what is the bias of the coin?

I am using the format from here Two players alternately flip a coin; what is the probability of winning by getting a head? except my equation looks 12/23 = p + (1-p)(11/23) and solving for p. I am getting p = 1/12.

I am not understanding the answer or if it is correct. If player A is more likely to win and has first flip, why is the chance of getting heads 1/12??

Assume the probability of heads is $p$. Now, let the probability that A wins be $P(A)$.

$$P(A) = p + (1-p)^2p + ...$$ $$\implies P(A) = \frac{p}{1 - (1-p)^2} = \frac{p}{2p - p^2} = \frac{1}{2-p}$$

Now, equating $P(A)$ to $\frac{12}{23}$, we get

$$2-p = \frac{23}{12} \implies p = \frac{1}{12}$$

So, yes, your answer seems correct. This method serves as a verification.

Now, to answer your question keep in mind that if $A$ goes first, it will have higher chance of winning no matter what the bias of the coin is.

• Can you explain how you went from the summation to 1-(1-p)^2? – user2202 Oct 28 '17 at 21:22
• Infinite GP sum – Siddhartha Oct 28 '17 at 21:24
• What is GP? I blieve it is summation but what does GP stand for? – user2202 Oct 29 '17 at 3:37
• Geometric progression, sorry I assumed you were familiar – Siddhartha Oct 29 '17 at 5:36